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Rewrite the expression in the form y^(n). Write the exponent as an integer, fraction, or an exact decimal (not a mixed number). root(4)(y^((7)/(3))y^((1)/(3)))=◻

Rewrite the expression in the form yn y^{n} .\newlineWrite the exponent as an integer, fraction, or an exact decimal (not a mixed number).\newliney73y134=□ \sqrt[4]{y^{\frac{7}{3}} y^{\frac{1}{3}}}=\square

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Full solution

Q. Rewrite the expression in the form yn y^{n} .\newlineWrite the exponent as an integer, fraction, or an exact decimal (not a mixed number).\newliney73y134=□ \sqrt[4]{y^{\frac{7}{3}} y^{\frac{1}{3}}}=\square
  1. Combine exponents of yy: Combine the exponents of yy by adding them, since they are being multiplied and have the same base.y(73)×y(13)=y(73+13)y^{(\frac{7}{3})} \times y^{(\frac{1}{3})} = y^{(\frac{7}{3} + \frac{1}{3})}
  2. Add exponents: Add the exponents (73)(\frac{7}{3}) and (13)(\frac{1}{3}).\newline(73)+(13)=(7+13)=83(\frac{7}{3}) + (\frac{1}{3}) = (\frac{7+1}{3}) = \frac{8}{3}
  3. Raise to fourth root: Now we have y83y^{\frac{8}{3}} under the fourth root, which means we need to raise y83y^{\frac{8}{3}} to the power of 14\frac{1}{4}.y834=(y83)14\sqrt[4]{y^{\frac{8}{3}}} = \left(y^{\frac{8}{3}}\right)^{\frac{1}{4}}
  4. Apply power rule: Apply the power rule for exponents (am)n=am∗n(a^{m})^{n} = a^{m*n}.(y83)14=y(83)∗(14)(y^{\frac{8}{3}})^{\frac{1}{4}} = y^{(\frac{8}{3})*(\frac{1}{4})}
  5. Multiply exponents: Multiply the exponents (83)(\frac{8}{3}) and (14)(\frac{1}{4}).83\frac{8}{3}∗\ast14\frac{1}{4} = 812\frac{8}{12} = 23\frac{2}{3}
  6. Write final expression: Write the final simplified expression. y23y^{\frac{2}{3}}

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